(see e.g. [1]}).
We also define the function \(\psi _0\colon M\rightarrow \mathbb {R}\)  by
\( {\left\lbrace \begin{array}{ll}-\Delta \psi _0 = - \kappa + \dfrac{2\pi \, \chi (M)}{\mathrm {Vol}(M)} \\\displaystyle \int _M \psi _0 \mathrm {vol}_g= 0\end{array}\right.}\) 
